黎曼几何

图书标准信息

• 作者 [葡]卡莫 著
• 出版社 世界图书出版公司
• 出版时间 2008-05
• 版次 1
• ISBN 9787506292184
• 定价 49.00元
• 装帧 平装
• 开本 24开
• 纸张 胶版纸
• 页数 300页
• 正文语种 英语
• 原版书名 Riemannian Geometry

【内容简介】

　　TheobjectofthisbookistofamiliarizethereaderwiththebasiclanguageofandsomefundamentaltheoremsinRiemannianGeometry.Toavoidreferringtopreviousknowledgeofdifferentiablemanifolds，weincludeChapter0，whichcontainsthoseconceptsandresultsondifferentiablemanifoldswhichareusedinanessentialwayintherestofthebook。
　　ThefirstfourchaptersofthebookpresentthebasicconceptsofRiemannianGeometry(Riemannianmetrics，Riemannianconnections，geodesicsandcurvature).AgoodpartofthestudyofRiemannianGeometryconsistsofunderstandingtherelationshipbetweengeodesicsandcurvature.Jacobifields，anessentialtoolforthisunderstanding，areintroducedinChapter5.InChapter6weintroducethesecondfundamentalformassociatedwithanisometricimmersion，andproveageneralizationoftheTheoremEgregiumofGauss.ThisallowsustorelatethenotionofcurvatureinRiemannianmanifoldstotheclassicalconceptofGaussiancurvatureforsurfaces。

【作者简介】

Manfredo Predigao Do Carmo，巴西国家纯数学和应用数学研究所教授。他所任职的研究所被认为是巴西数学领域的数学研究与教育机构。

【目录】

Prefacetothefirstedition
Prefacetothesecondedition
PrefacetotheEnglishedition
Howtousethisbook
CHAPTER0-DIFFERENTIABLEMANIFOLDS
1.Introduction
2.Differentiablemanifolds；tangentspace
3.Immersionsandembeddings；examples
4.Otherexamplesofmanifolds，Orientation
5.Vectorfields；brackets，Topologyofmanifolds

CHAPTER1-RIEMANNIANMETRICS
1.Introduction
2.RiemannianMetrics

CHAPTER2-AFFINECONNECTIONS；RIEMANNIANCONNECTIONS
1.Introduction
2.Affineconnections
3.Riemannianconnections

CHAPTER3-GEODESICS；CONVEXNEIGHBORHOODS
1．Introduction
2．Thegeodesicflow
3．Minimizingpropertiesofgeodesics
4．Convexneighborhoods

CHAPTER4-CURVATURE
1．Introduction
2．Curvature
3．Sectionalcurvature
4．Riccicurvatureand8calarcurvature
5．Tensors0nRiemannianmanifoids

CHAPTER5-JACOBIFIELDS
1．Introduction
2．TheJacobiequation
3．Conjugatepoints

CHAPTER6-ISOMETRICIMMERSl0NS
1．Introduction．
2．Thesecondfundamentalform
3．Thefundarnentalequations

CHAPTER7-COMPLETEMANIFoLDS；HOPF-RINOWANDHADAMARDTHEOREMS
1．Introduction．
2．Completemanifolds；Hopf-RinowTheorem．
3．TheTheoremofHadamazd．

CHAPTER8-SPACES0FCONSTANTCURVATURE
1．Introduction
2．TheoremofCartanonthedeterminationofthemetricbymebnsofthe　curvature．
3．Hyperbolicspace
4．Spaceforms
5．Isometriesofthehyperbolicspace；TheoremofLiouville

CHAPTER9一VARIATl0NS0FENERGY
1．Introduction．
2．Formulasforthefirstandsecondvariationsofenezgy
3．ThetheoremsofBonnet—MyersandofSynge-WeipJtein

CHAPTER10-THERAUCHCOMPARISONTHEOREM
1．Introduction
2．TtleTheoremofRauch．
3．ApplicationsoftheIndexLemmatoimmersions
4．FocalpointsandanextensionofRauch’sTheorem

CHAPTER11—THEMORSElNDEXTHEOREM
1．Introduction
2.TheIndexTheorem

CHAPTER12-THEFUNDAMENTALGROUPOFMANIFOLDS0FNEGATIVECURVATURE
1．Introduction
2．Existenceofclosedgeodesics
CHAPTER13-THESPHERETHEOREM
References
Index